Definition:Inverse Relation

Definition

Let $\mathcal R \subseteq S \times T$ be a relation.

The inverse relation to (or of) $\mathcal R$ is defined as:

$\mathcal R^{-1} := \left\{{\left({t, s}\right): \left({s, t}\right) \in \mathcal R}\right\}$

That is, $\mathcal R^{-1} \subseteq T \times S$ is the relation which satisfies:

$\forall s \in S: \forall t \in T: \left({t, s}\right) \in \mathcal R^{-1} \iff \left({s, t}\right) \in \mathcal R$

Domain and Range of Inverse Relation

Note that the preimage of a relation is the image of its inverse, and vice versa:

• $\operatorname{Im}^{-1} \left({\mathcal R}\right) = \operatorname{Im} \left({\mathcal R^{-1}}\right)$
• $\operatorname{Im} \left({\mathcal R}\right) = \operatorname{Im}^{-1} \left({\mathcal R^{-1}}\right)$

Also known as

An inverse relation is also seen as converse relation.

Some sources use the notation $\mathcal R^\gets$ instead of $\mathcal R^{-1}$.

Others use $\mathcal R^t$, for example T.S. Blyth: Set Theory and Abstract Algebra (1975).

Others use $\breve {\mathcal R}$, for example Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (1936).

Also see

• Inverse of a Mapping
• Inverse Mapping

• Results about inverse relations can be found here.

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 10$: Inverses and Composites
• Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• Seth Warner: Modern Algebra (1965)... (previous)... (next): Exercise $5.8$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 14$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$: Problem $\text{AA}$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 4$: Exercise $4$